**Department of
Mathematics**

**UNIVERSITY OF
ROCHESTER**

**Rochester, New
York 14627**

**Tel (585)
275-4429 or 244-9368**

**FAX (585)
273-4655 (at U of R)**

**email:
RARM@math.rochester.edu**

*Ralph A. Raimi*

*Professor Emeritus*

** 25 May 2002**

**David Garbasz**

**12 West 72nd Street**

**New York, NY 10023**

** **

Dear Mr. Garbasz:

Here
is the report you requested. I have
written it as if in my own

hand, which it largely is so far as
prose is concerned, though it is based on a

draft first written by Braden
alone. We have had to confer on the
grades of

course. Let me mention one curious feature of the grading.

When Braden and I first evaluated state
standards for the Fordham

Foundation we gave separate scores of 0
to 4 for each of our four criteria,

never thinking to average them or sum
them. The Criteria (Clarity, Content,

Reason, Negative Qualities) were of
different dimensions, and we expected

the readers to consider them as
such. However, at the time of our study

there were also evaluations of
standards being written by the American

Federation of Teachers and by the
Council for Basic Education, each of

which was giving a summary grade of A,
B, C and so on for each state.

Chester Finn, President of the Fordham
Foundation, found himself obliged

to do the same with the reports he
commissioned; also, there were many

potential users of the report (State
Governors, for example) who needed an

overall grade for local purposes. Braden and I therefore summed the four

separate scores, as you will see below,
and created a grading scale out of thin

air, averaging apples and oranges as it
were. The reader of this report on

Israel's Curriculum 2000 should take
note of this warning, and pay more

attention to the numbers taken
separately for our four criteria of judgment

than to the summed score, and more
attention to the commentary than to

the numerical or alphabetical scores.

__Part 1, Criteria and Evaluation Results__

The
authors of this report are Lawrence Braden, a high school

mathematics teacher at St. Paul's
School in Concord, New Hampshire, and

Ralph A. Raimi, Professor Emeritus of
Mathematics at the University of

Rochester in Rochester, New York. The
words printed here are mainly those

of Raimi, who has put them into a letter
to David Garbasz, a mathematician

living in New York City. We have been invited by Mr. Garbasz to write
an

evaluation of the current draft of the
Curriculum 200 (Mathematics

Curriculum), as proposed by a committee
appointed by the Department for

Curricula (of Israel's Ministry of
Education), to serve as a replacement for a

1998 document having similar
purpose. We were provided with an
English

translation of the proposed Curriculum
2000 by Mr. Garbasz, who has said

he wants us to evaluate, for his use
and for transmission to interested persons

in Israel, this Israeli
"standards" document using the same criteria we had

earlier used for two such studies we
made for the Thomas B. Fordham

Foundation of Washington, D.C.,
published in 1998 and 2000. In those

reports we made comparative judgments
of the "Standards" for school (K-12)

mathematics education as they had been
published (separately) by each state

of the United States in the late 1990s
at the urging of the federal government;

and we included a corresponding
Japanese "standards" for comparison as

well.
These American reports can be found on the web site of the Fordham

Foundation, at
<http://www.edexcellence.net/standards/math.html> and at

<http://www.edexcellence.net/library/soss2000/2000soss.html>,
respectively.

(As
to the qualifications of the authors of these two Fordham reports,

and of our report on Curriculum 2000
contained below, we can summarize

briefly as follows: Ralph Raimi has
been a professor of mathematics at the

University of Rochester since 1952,
coming from having been a student and

teaching assistant in mathematics at
the University of Michigan, from which

he holds degrees in physics (B.S.) and
mathematics (PhD). In earlier years

he served as a communications and radar
maintenance officer in the U.S.

Army Air Forces during World War
II. Lawrence Braden has had a long

career as a teacher of mathematics in
both public and private schools, in

Hawaii, Russia and New Hampshire, at
elementary, junior high school and

high school levels, and with classes of
various social dimensions, from poor

students with limited English
proficiency to the college-bound children of the

affluent. In 1987 Mr. Braden was invited to Washington to receive a Presi-

dential Award for Excellence in
Mathematics and Science Teaching.)

The
second of the Fordham reports is brief, being a revision of the earlier

reports in that it evaluated only those
states which had made some

changes in the two-year interval. It is the first of our reports, State

Mathematics Standards: An Appraisal of
Math Standards in 46 States, the

District of Columbia, and Japan (1998),
to which we wish to call the readers'

attention, because that volume contains
a lengthy exposition of the Criteria

we used in our evaluation, criteria
which are only briefly summarized in the

second (2000) report and which will
also be rather briefly summarized

below.
While the full Fordham exposition of our criteria contains much

information on what we consider
important in school mathematics

education, it is strictly speaking not
necessary to consult it order to

understand the meaning of our
evaluations given below; but it might be

useful to one who wishes to understand
why we have given the weights we

did to the various aspects of the
documents studied. The present report

does not contain a full, detailed
argument for the scores we awarded, and the

quotations we present below, and
comment upon, are only a sampling of

what could be adduced for the
purpose. Even so, this report is
considerably

more detailed than what we provided for
each of the American states in 1998

and 2000, there being so many of them.

In
the present report, then, we will do the same for the Israeli

Curriculum 2000, hereinafter to be
referred to as "Curriculum 2000" (or,

sometimes, "the document"),
as we did for the American state standards,

using the identical criteria, which
fall under four headings:

I. Clarity refers
to the success the document has in conveying to the reader

its own purpose: What it considers important, necessary,
testable, useful and

so on.
The organization of the document and the quality of its prose are

evaluated under this heading, without
regard to the value of the advice

actually given, or the relevance of the
curricular items themselves. In our

Fordham reports we subdivided this
heading into three subheadings,

(a) Clarity of language;

(b) Definiteness of the prescriptions; and

(c) Testability of the recommended or prescribed
content.

We will use the same subheadings here,
giving something between 0 and 4

points at each subheading and averaging
the three results to obtain a maximal

score of 4 points for the entire
category named Clarity, 4 being excellent or

exemplary, and 0 a bad failure. We recognize that some of the pedagogical

recommendations of Curriculum 2000 are
not "testable" in the nature of

things; this is also the case with some
of the American frameworks, and

nobody is downgraded on this
account. It is the outcomes demanded,
the

skills asked for, that we wish to be so
plainly described that it is possible for

someone other than the teacher in the
classroom to determine whether they

have been conveyed.

II. Content refers to the curricular items
themselves: Does the document

ask what should be asked of students at
the grade levels in question, in such

a way as to conduce to a coherent
program containing the information and

developing the skills we considered
optimal. In our Fordham document we

used three subheadings, one each for
primary, middle and high school levels.

In the present case, therefore, our
evaluation of the Curriculum 2000 is not

quite comparable with our American
states' (and the Japanese) evaluations,

as Curriculum 2000 is concerned only
with the primary level and its score

will not be an average of three
scores. (In some of the Fordham
evaluations

a poor "primary grades" score
might have been somewhat balanced by a high

"high school" content score,
or vice versa, something not possible here.)
The

reader should make proper allowances
for this difference in interpreting our

overall scoring, which sums the results
of the four separate criterion scores.

III. Reason refers
to the structural organization by which the parts of

mathematics cohere as an intellectual
system. Even from the earliest grades

it is necessary that the student
recognize the status of his own knowledge:

what is empirically known, what is
approximate, what is exact, what is

accepted (perhaps for the time being
only) on authority, and what is

deducible from something else.
Reasoning is also required in the

mathematical modeling of real
phenomena, but the model is futile if the

mathematical properties of the model
are not understood. While the grade

level of Curriculum 2000 is elementary,
so that formal deductive reasoning

exercises and skills cannot be expected
to appear as they should in later

grades' algebra and geometry, for
example, we did expect the groundwork for

such reasoning, and the appreciation of
the connectedness of the subject

matter, to be inculcated from the
beginning. And indeed, those American

states whose later years' demands
avoided the deductive essence of

mathematics were in fact also those
whose programs for earlier years were

visibly lacking in the corresponding
qualities.

IV. Negative
Qualities refers to two common classes of failings of attitude or

philosophical stance, which we labeled

(a)
False doctrine; and

(b)
Inflation.

By
false doctrine we mean either outright mathematical error or

"advice which, if followed, will
subvert instruction in the material otherwise

demanded" (or which otherwise
should be demanded). Included are the

excessive reliance on calculators,
excessive insistence on "real-world"

application as the criterion for value
in mathematical instruction, and the

unreasonable delay (on constructivist
principles) in offering direct information.

Such prescriptions of delay, setting
children adrift to guess and

argue with little direction, habituates
students to avoid the learning they might

have accomplished in the same time by
taking advantage of the efforts of

their more learned ancestors.

By
inflation we mean "language whose only purpose is to fill paper, or

to suggest a profundity impossible for
the level in question..." Such
language

weakens the respect the reader or user
of the document should have for its

advice even in places where well
taken.

In
scoring Criterion IV we use a reverse scale, 4 points being

awarded for the absence of the quality
(false doctrine or inflation) in

question, so that the grade for this
criterion can be added to those of the

other three in arriving at a total.

In
the scoring under each criterion, "4" need not require perfection,

and "0" need not require
vacuity; they are merely rankings of quality.
The

grade of A, B, C, D, or F (for
"failure") is then awarded to the document as a

whole, based on a score from 0 to 16
obtained by summing the scores under

the four criterion headings. In our Fordham Foundation reports, the grade

of A was awarded for scores of 13 or
more, B for scores of 11 or more, but

less than 13, C for scores of 7 or
more, but less than 11, D for scores of 4 or

more, but less than 7, and F for scores
of less than 4. We do the same here.

For
comparison with the score of Curriculum 2000, we should

mention that by these criteria, Japan,
California, North Carolina and Ohio

received grades of A in 1998, nine
states received a grade of B, seven states a

grade of C, twelve states a grade of D,
sixteen states a grade of F, with four

states not judged for various
reasons. By 2000, the second Fordham
report

found some improvement, and six states
were awarded an A (not counting

Japan, which had not changed its
document in the interim), while only eight

received an F, with two states not
graded.

Our
grades for Curriculum 2000 are as follows:

I.
Clarity....................................……………...… 3 points

(a)
Clarity of language ........ 4 points

(b)
Definiteness .............…. 2 points

(c)
Testability ............……. 3 points

II. Content
......................................………….... 1 point

III. Reason
...................................…………...... 1 point

IV. Negative
qualities...........................……....... 1.5 points

(a)
False Doctrine .............. 0 points

(b)
Inflation ................…... 3 points

The
total score, 6.5 points out of a possible 16, rates an overall grade

of D on the Fordham Foundation scale,
though not far from the lower

boundary of C. It is worth noting that most of the
document's points are the

result of good language and an
avoidance, by and large, of the bureaucratic

or pretentious language that infests so
many American pronouncements in

education. It is possible that in 1998 we awarded too many points for this

part of our assessment for reasons that
were rather parochial and ought not

to be applied internationally, but we
are here committed to providing a grade

comparable to those we gave then. Amusingly, one might notice that a blank

piece of paper submitted as a
"Standards" or "Curriculum" would already have

four points, hence a grade of D, merely
for absence of negative qualities.

__Part 2, Commentary on the Evaluation__

Curriculum
2000 is a rather clearly written document of almost two

hundred pages (in our English
translation), plus an introductory essay of

another 19 pages setting forth the
intentions of the Planning Committee

which prepared it. The philosophy of the Committee are
certainly made

plain, and it is exemplified in what
follows. Having studied the entire

document, we have found that it would
have been almost sufficient to cite

phrases of this introduction alone in
support of our conclusions, and

convenient as well, since the entire
Curriculum 2000 is too long to admit a

full treatment item by item here. We did study the entire document,

however, and will submit some quotations
from the main body of

Curriculum 2000 when needed.

After
the introduction, each of the six Grades is given a parallel

treatment: A short essay headed __An Approach to Mathematics Teaching in __

__the Xth Grade__; a __Table
of Topics__ for each of the major divisions addressed

in each grade: Number and Operations,
Geometry and Measurement, and

Data Analysis; and a section called __Details
and Exa__mples. In every grade the

document recommends to the teacher (and
presumably to the textbook

writer) a ratio of 65:25:10 for the
percentages of time given to the three

major divisions (Number, geometry,
data).

Each
__Table of Topics__ has subdivisions of several sorts in each grade,

and the items are sometimes rather
summary, but then clarified further in the

third table associated with each
Grade: the Details and Examples, which
are

given in two columns thus
headlined. The examples are often
typical

problems to be solved, or activities to
be engaged in over some period of

time, though in the document we
reviewed there was some empty space

where other examples could have found a
place.

There
is in all a good deal of overlap from one grade to another, but

it is harmless and indeed necessary if
each Grade is to stand on its own, for

some readers might well consult the
document for one particular grade of

interest without looking closely at the
rest of the document. However, in

reading the whole it is not always
apparent where the progression of ideas is

to be found, from the simpler to the
more complex, from the basics to that

which is built on the basics. Which topics are more important than
others?

If calculators are an "inseparable
part of the activities" in lessons designed to

"help the children develop a sense
of self-confidence in their ability to handle

mathematics", what emphasis should
be given to things that must be done in

the absence of the calculator? The insistence on calculators begins with
the

first grade: "It is important to integrate the use of calculators
throughout the

year and in daily life in parallel with
developing skills of calculation orally, in

writing and in estimating." We do not believe this, and this and similar
sen-

timents are repeated in nearly
identical terms in all succeeding grades.
One

would think that
"self-confidence" would result from the internalizing of what

the calculator should no longer be
needed for, as time goes on.

On
the other hand, the standard algorithms of arithmetic and the

uses of memory are downplayed when not
denigrated, and much space is

given in Curriculum 2000 to explaining
what need not be learned or

considered at this or that stage, as if
the authors were fearful of too much

learning. In the Introduction is written, "Studying algorithms is
important to

the extent that they support the
students' conceptual understanding of

numbers and their operations. As a result, when studying topics that are

related to numbers and operations,
there is no need to focus on either

calculations or the study of
traditional algorithms."

This
statement, which we counted as a major "false doctrine", is

similar to many that are put forward in
the United States, and which have led

to the utter abandonment of all
arithmetic algorithms in some modern

programs that are avowedly in concord
with the recommendations of the

(American) __National Council of
Teachers of Mathematics__, but one cannot

condemn a statement on the mere grounds
that its misinterpretation would

lead to evil results. However, the statement as given in
Curriculum 2000 all

but begs for such misinterpretation,
for it does not say just to what extent the

traditional algorithms do or do not
support the "conceptual understanding",

and it does not define the intensity of
"focus", especially when the focus

warned against is on more than one
thing.

As
to the algorithms' "support ... [of] conceptual understanding",

Curriculum 2000 exhibits in its
specific recommendations a belief that they

contribute very little, since very
little in the way of algorithm is required --

indeed, they are mostly deplored -- and
"student-constructed" algorithms, or

devices, are forever celebrated over
the old ones besides. And to say that

one does not "focus" on
calculations and algorithms is -- at the very least -- to

say the very least. It will not be misinterpretation that will
cause followers of

Curriculum 2000 to scant calculation,
both of decimal arithmetic and

fractions, to the detriment of the
students' mathematical future; it will be a

correct apprehension of the import of
the objectives as announced in that

document.

At
the 6th Grade level, the Approach section states something that is

in our view diagnostic of the entire
philosophy: "The program sets goals and

limits [our emphases] for the performance of calculative
algorithms in the 6th

grade ... In the division of common
fractions, it is sufficient that problems are

solved at the level of insight
(dividing a fraction by an integer) without

demanding the division algorithm (changing
the division into multiplication

by the inverse number). In the division of decimal numbers, the
solving of

xercises based on immediate division is
enough (for instance, 0.8 : 0.4)

without demanding the division algorithm..."

Yet
back in Grade 5 the student was asked, albeit with fractions

whose denominators are not greater than
12, to understand "the significance

of a fraction as a quotient of
division", and to handle common denominators.

Why then the limitation?
What is the student supposed to do with

common denominators if the arithmetic
of fractions is to be so limited? It is

as if "invert and multiply"
were inevitably a mindless algorithm, stunting the

student's growth, yet an important use
of common denominators is one way

to make the algorithm reasonable and
memorable. It may well have been

the case that this famous algorithm was
mindlessly prescribed in the bad

teaching we have all been acquainted
with, but the cure for bad teaching is

not the withholding of information,
information which is absolutely essential

if the student is to have any hope of
mastering any part of algebra two or

three years later, where the
manipulation of fractions is a notoriously difficult

pedagogical problem in the best of
times. The teaching of "invert and

multiply" is not simply the
presentation of a tool the output of which is the

"answer"; it is part and
parcel of the meaning of fractions, and the whole

rational number system. Furthermore, the "self-confidence"
generated by

limiting denominators to integers below
13 will soon evaporate when the

student finds he has been cheated in
his earlier years, and that the principles

involved in acquaintance with fractions
of all possible denominators,

principles that have been denied to
him, principles which were the glory of

the development of arithmetic in
Renaissance Europe, and were not

discovered by children, are more
important than the decimal approximations

put out by a calculator.

Furthermore,
"mindlessness" of a sort is a necessary feature of

education and civilized behavior. We do not reduce our judgments to first

principles every time we have a
decision to make; we must make use of

memory, the accumulation of things we
have thought about and settled in the

past, and have made into habit. Otherwise we waste our lives in repeating

the past, painfully recapitulating both
our own earlier labors and the work of

our ancestors. It is the antithesis of education to teach
children not to

remember and freely use earlier results
(often called algorithms) in favor of

forever starting anew.

Number
and calculation is quite properly the major concern of the

early grades, with some geometry in
connection with measurements such as

everyone must be able to make. To devote 25% of the program to geometry

and measurement is not too much, if
wisely done, though we consider the

10% allocated to "data
analysis" excessive, since its content is so little. Even

the average, or mean, which could be
introduced as soon as division is

studied, is not demanded as an item of
data analysis until Grade 5, though

students are encouraged to invent some
of their own measures of dispersion

of mean, in addition to
"median" and "range", which are provided by the

curriculum. Instead, Grade 4 suggests, under the Data Analysis heading,

"Whole research projects: short
surveys and projects that continue for several

lessons. For example: “What do the
fourth graders do after school? ; Do

cars stop at the stop sign in our
neighborhood? How can water be

conserved in school?" Mathematics
class time is precious; these valuable

researches should be reserved for
"social science", perhaps.

The
geometry of Curriculum 2000 is much too primitive to deserve

their 25% even for these grade levels,
for they amount mainly to exercises in

nomenclature, with the occasional
warning that certain things are too difficult

to include, and sometimes the avoidance
of mention of the status of some

piece of information. A 5th grade "Investigation"
begins, "It is given that the

sum of the angles of a triangle is 180
degrees. An example for finding the

sum of the angles of a hexagon: We divide the hexagon into triangles by
..."

To us, and we suppose to the student or
teacher as well, the status of the

remark "it is given that the sum
of the angles of a triangle is 180 degrees" is

much like "Let a polygon have five
sides..." It sounds like an
assumption,

which could have been otherwise. Now it might well be that we don't expect

a Euclidean proof of this property of
every triangle, but the suppression of

the status of this piece of information
is certainly at odds with the ambition of

the "investigation" that
follows, concerning the angles of a hexagon.

Informal
exercises in visualization are necessary in their place, and

can be pleasant games as well. But a simple lesson once learned need not be

belabored for years, or even minutes,
of further instruction. One fifth
grade

exercise, to discover whether it is
possible for a parallelogram can have a

diagonal which has the same length as
its "height", is rather clever, even if

"height" is an idea having as
much to do with the placement of the

parallelogram in the ambient
gravitational field as with its geometric

properties. But this is an artificial linguistic question, a puzzle only
because

of our predisposition to regard
"height" as the shorter of the two possible

heights, since placing the
parallelogram in the other position makes it seem a

bit unstable, gravitationally; so that
textbooks seldom print pictures of "tall"

parallelograms. The student is in this exercise trapped by a
word, an artifact

of the classroom rather than of
geometry; and if he discovers the common

schoolroom misconception the words have
led him into he has not thereby

really learned something about
parallelograms or geometry. Such a
student

can be easily taught that a
parallelogram has two possible "altitudes", as a

triangle has three, and will have
missed nothing by his "research" into the

possibility of the anomalous
"diagonal" of this particular exercise. This is not

to say the problem is a bad one, only
that by making so much of "discovery",

as the constructivist philosophy
popular in the United States does, there is

time wasted that could have been put to
better use. The idea that students

are imitating professionals in
"researching" some puzzle of this sort is in fact

generally mistaken, for the first thing
professionals do is to find out whether

the answer has already been found by
someone else. Real research proceeds

from a supply of previous information,
not by referring to the definitions, or

the intuitions, at every stage.

And
while all this is going on there is real geometry that is not being

attacked. It might be early, in Grade 6, to expect demonstrative geometry
in

all its formality, but there is not
even a hint in Curriculum 2000 that any such

thing is in the offing in the study of
geometry. For example, the definition
of

a circle. It is pointed out that all points of a circle (once drawn) are

equidistant from a point called the
center. Somewhere else it is noted that

there is a radius. But if students following this curriculum
were asked to

define the word "circle" they
might well say it is "all points equidistant from

the center". We have seen college students who say such
things. Yet

accuracy of definition is not beyond
the power of elementary school

children, or would not be if they were
exposed to careful definitions, and

required to discuss and find just where
a poor definition falls short of what

they already have intuition of. Instead, Curriculum 2000 says things like,

"Students must talk about the math
they are learning... They must document

their solutions methods, their assumptions,
... other possible solutions."

Good advice for those who have mastered
the language, but in Grade 5, for

which this behavior is prescribed, such
talk has proved unfruitful in many an

American classroom. Accurate writing and speaking have to be
taught, and

will not be learned by students
listening to one another. Curriculum
2000

has few if any mentions to this effect,
and this failure contributes to the poor

score we have given it under the
criterion called Reason.

In
the __Table of Topics__ for Grade 6, the rubric __Mathematical Insight__,

opposite the topics
"proportion" and "finding the ratio", is the instruction,

"Non-formal experience of directly
proportional situations." There
follows a

blizzard of more detailed suggestions
in this connection, including

manipulatives such as "nail
boards" and "strips to represent lengths" -- in

Grade 6! But despite the details, the key words have already been
written:

"non-formal",
"experience", and "situations". The student is required not to

know too much, and to take what he does
know from experience in the real

world of nail boards rather than in
reasoning, for example. In working with

proportionality one does not expect him
to recite "There exists some k>0

such that y = kx" at this level,
but the answer is not to consume the day, or

days, with projects illustrating
proportional relationships, some of them

doubtless having to be drawn, or built,
or cut out of the newspaper (which is

what happens in America these days),
while a great ocean of mysteries

concerning fractions lies all before
him. Why not give him a little help?

The
students are asked somewhere in Grade 6 to obtain decimal

equivalents to such fractions as 1/2
and 3/4, but the mental strain of sys-

tematically finding (even for simple
examples) the "whole" when a given

percentage is known is considered
beyond their powers: "Finding the
whole

quantity from a given percentage (not
necessarily by building an exercise of

calculation)" is the instruction
as it appears in the translation we have.
Why

not a calculation? If 3/5 (or 60%) of a number N is given
surely N is 5/3 the

given number. And even if it is a
complicated percentage such as 61% (i.e.

61/100) of N that is known, then N is
as surely 100/61 times the known

number. Calculators, by the way, have nothing to do with this
understanding,

which ought to be commonplace even in
the 5th grade, else what are

fractions for? And surely by the 6th grade one need no
longer buy toys to

illustrate fractions, even if so urged
by publishing houses.

In
another place (5th Grade Table of Topics for geometry) it is

bravely stated that the student should
know the difference between a

statement and its converse, but the
entire Curriculum 2000 contains nothing

resembling a theorem. Again, here the advertising is excessive,
and

constitutes inflation by the rubrics of
our evaluation. Theorems do abound

at the elementary level, though they
are not usually construed as such.
Every

algorithm is a theorem, stating that a
certain definite procedure will infallibly

yield a certain definite result. The intellectual value of these algorithms
and

the analysis that shows them to be
correct far outweigh the supposed benefits

of the use of the algorithms in daily
life. Every adult will of course use a

calculator when faced with an intricate
calculation of practical use, but it does

not follow that children should be
taught to do the same, especially with

simple calculations such as appear in
Grades 1 and 2, where Curriculum

2000 prescribes calculators as an
inseparable part of the curriculum.
There

are many things adults do in daily life
that we do not prescribe for children,

for the process of learning is an
artificial one, not to be confused with the

experiences of daily life (though of
course the applications should also be

taught when possible).

Some
things really have nothing to do with "experience" and

"situations", and are not
even understandable when only given "informally",

but are stepping-stones to other things
in real life. Arpeggios are not real

music, but are essential in learning
real music. For ages in America
children

have built vocabulary and reading
ability with the aid of "spelling bees", than

which nothing could be more
artificial. Knowing the algorithm by
which the

fraction 11111/13 can be converted to a
decimal appears impractical (our

calculator gives the answer
"854.69230770", from which one would be hard

put to deduce any periodicity of the
digits in the long run); yet the fact that

any such quotient is ultimately
periodic is something students in the high

schools will be asked to understand --
unless the high school mathematics

curriculum is by then similarly
disfigured by the insistence on calculators as a

path to "understanding". It is the apparently useless algorithm that in
the long

run will yield up a periodicity; and it
is furthermore a theorem that, conversely,

a periodic infinite decimal expansion
represents a fraction. Both theorems

are fairly profound, and not suited to
the 6th grade curriculum, but the understanding

and skill at elementary calculation
must be achieved by

that stage if the later theorems are
ever to be accessible. It is not fair
to

children to cut off their later
development by building a false self-confidence

based on failure to offer challenges.

One
more example, where Curriculum 2000 deliberately scants the

prefiguring of important later
developments, occurs in the discussions of area

and volume. Volume is rather informally understood as something

measurable by the pouring of liquids,
and area as something to be derived

when possible from tilings, though it
is hard to trace the development of the

idea of area in Curriculum 2000, apart
from the fact that finite partitions of

polygons preserve the original area
when reassembled. (Much is made of

the tangram, for some reason.) At any rate, here are extracts from the 6th

grade Details and Examples for
geometry, concerning properties of circles:

"The
connection between the diameter and the circumference of a

circle: This investigative activity can be performed in various
ways. Its aim is

to reach the following
conclusions:

One. There is a connection between the diameter
of a circle and its

circumference. This connection exists in every circle,
large and small.

Two. The ratio of the diameter to the
circumference is equal to

nearly 3. There is no need to reach the exact formula."

"There
are various ways of measuring the approximate area of a circle

(the method of measuring area by units
of tiles is not suitable for a circle).”

"The
area of a circle equals approximately the area of three squares

whose sides are equal to the radius of
the circle. There is no need to reach

the exact formula."

In
its anxiety to forestall the mere "memorization" of formulas, the

program prescribes "investigative
activities" resulting in formulas guaranteed

to be informal and incorrect, all the
while warning against "exact" formulas.

There is neither practical value nor
intellectual value in such uses of time.

But time-wasting is not the only evil
in this passage, since "the method of

measuring area by units of tiles"
is not a mere measuring device; it is the very

definition of area. Its importance is enormous, and in principle
it is the only

method of measuring the area of a
circle. It would be better to leave off
all

these geometric activities and
adventures in favor of some straightforward

information, with verification and
experience as needed over time. The

same must be said, even more
forcefully, about the arithmetic.

The
sections on Data Analysis fortunately are scheduled to take up

only 10% of instructional time, for
they are quite empty of content.
Students

learn what the median is, and can spend
a lot of time collecting numbers and

finding the median and range. When they are old enough to add some

numbers and divide (with a calculator)
by the size of the sample, they are

permitted to add "average" to
their vocabulary. It is not possible to
see what

else there is in this 10% of the first
six years of mathematics prescribed by

Curriculum 2000, except for some
graphical displays and "projects", with the

intellectual content no more than can
be accomplished in a week or two in

middle school or high school, where it
can be combined with some

probability considerations.

All
told, then, we give Curriculum 2000 a more or less failing grade.

We should mention here that our
instructions were to grade Curriculum

2000 according to the criteria we used
in our two Fordham reports. This

means that large parts of Curriculum
2000 have had to be ignored, since they

refer to pedagogy alone, something that
forms part of a good number of

American states'
"frameworks", which Curriculum 2000 resembles more than

the bare-bones "Standards"
which generally form only part of a framework in

states that publish in this form. Yet the pedagogical advice is not devoid of

information concerning the intent of
the standards part of the document, for

it permits one to deduce something of
the motivation of the authors.

Motivation
is always dangerous to attempt to fathom, especially the

motivation behind something one
disapproves of. In the case of
Curriculum

2000, however, some of the motivation
is expressly stated, and so it is safe to

mention it. In the Introduction it is
written that "The committee once again

stresses that the main goal of the
study of mathematics in the elementary

school is to avoid failure, and build
each young student's confidence that he

or she can study math with enjoyment
and success."

It
is possible that the translation into English is imperfect here, for

surely avoidance of failure and the
building of confidence cannot be the

main goal, but even with modification
this statement is an announcement of

values we see richly exemplified in the
sequel. The American NCTM

(National Council of Teachers of Mathematics)
has put forward such a

position beginning in 1980, culminating
in its Standards of 1989 which have

had a great impact on American schools,
and evidently abroad as well. The

NCTM has retreated somewhat, though
only somewhat, in its second edition

of the Standards, now called PSSM
(Principles and Standards for School

Mathematics), published in 2000. Meanwhile, in America, there has been a

considerable reaction to the
"dumbing down" of the curriculum, whose

purpose is the avoidance of the announcement
of failure and the building of

self-esteem, and that reaction has now
garnered several years of experience

in California, where an avowedly
anti-NCTM framework has been adopted,

and corresponding textbook selections,
and the results have been quite the

opposite of the predictions of
widespread failure and despair.

Much
depends on the quality of teaching, of course, and there are

environmental factors (common to many
other countries as well) that are still

insuperable whatever the curriculum
might be, at least for some students; but

the removal of content from the
curriculum is only a hiding of failure.
One

does not "avoid failure" by
simply avoiding all challenge that could possible

lead to failure; one merely avoids its
announcement. And one does not really

build self-confidence with constant
praise; a student soon learns that there

has to be something to be confident
about, and indeed comes to suspect all

praise, even if deserved, since
incessant praise and avoidance of "judgment"

have given him no standard of value.

We
see Curriculum 2000 as a long step in the direction that our own

NCTM took twenty years ago, with no
success. America has not risen in

international comparisons in
mathematics, its students have still been failing

(though sometimes the failure is
concealed under other titles) in the usual

numbers, and its "successful"
students, who are the future of its technology

and its general intellectual life, are
coming to college with less and less

necessary background all the time. Our technical industries are

consequently becoming the employers of
immigrants from the Far East,

Russia and India, where early
mathematical education is the antithesis of

what NCTM promulgates. And it is not only technology, or science,
or

mathematics, that is being badly served
by this conversion of mathematics

instruction into happy games, but the
whole of our culture, of which

mathematics is an inseparable part.

It
is not possible for professional mathematicians to "take over" the

teaching of mathematics in the
schools. If they could, or did, they
would

cease to be mathematicians; that is not
their trade. Some few have taken an

interest in the matter, but their
advice is generally rebuffed by those in the

profession of teaching mathematics or
training the teachers of school

mathematics, professors in teachers
colleges and so on, on the grounds that

as mathematicians they don't understand
the nature of pedagogy or its

problems. And it is probable that the average mathematician would not be

successful if drafted to teach at the
5th grade level, or at least would not be

comfortable there, and might therefore
not work with the proper enthusiasm.

But it does not follow that he does not know what can be learned at that

level, and ought to; his experience as
a mathematician will give him an insight

into the ordering of topics, the
distinctions between relevant and irrelevant

subject matter, the purpose of
sometimes obscure relationships and

notations, that are not fully
appreciated by those proposing curricula and

examinations for the schools.

If
we can make any single proposal to a committee seeking to write a

mathematics curriculum, in any country
in the world today, including the

United States, and whether at elementary
or advanced level, it would have to

be this: Make sure the next round of writing is conducted with the active

participation (and not just editing or
after-the-fact criticism) of as many

mathematicians as of professional
school educators.

This
recommendation is not only the recommendation of Ralph Raimi,

the mathematician, but equally that of
Lawrence Braden, the teacher.

It is plainly not the advice the
Ministry of Education in Israel has received

when preparing Curriculum 2000.

Sincerely
yours,

Ralph
A. Raimi

Lawrence
Braden